Mathematics for the interested outsider

(Pseudo-)Riemannian Metrics

Ironically, in order to tie what we’ve been doing back to more familiar material, we actually have to introduce more structure. It’s sort of astonishing in retrospect how much structure comes along with the most basic, intuitive cases, or how much we can do before even using that structure.

In particular, we need to introduce something called a “Riemannian metric”, which will move us into the realm of differential geometry instead of just topology. Everything up until this point has been concerned with manifolds as “shapes”, but we haven’t really had any sense of “size” or “angle” or anything else we could measure. Having these notions — and asking that they be preserved — is the difference between geometry and topology.

Anyway, a Riemannian metric on a manifold is nothing more than a certain kind of tensor field of type on . At each point , the field gives us a tensor:

We can interpret this as a bilinear function which takes in two vectors and spits out a number . That is, is a bilinear form on the space of tangent vectors at .

So, what makes into a Riemannian metric? We now add the assumption that is not just a bilinear form, but that it’s an inner product. That is, is symmetric, nondegenerate, and positive-definite. We can let the last condition slip a bit, in which case we call a “pseudo-Riemannian metric”. When equipped with a metric, we call a “(pseudo-)Riemannian manifold”.

It’s common to also say “Riemannian” in the case of negative-definite metrics, since there’s little difference between the cases of signature and . Another common special case is that of a “Lorentzian” metric, which is signature or .

As we might expect, is called a metric because it lets us measure things. Specifically, since is an inner product it gives us notions of the length and angle for tangent vectors at . We must be careful here; we do not yet have a way of measuring distances between points on the manifold itself. The metric only tells us about the lengths of tangent vectors; it is not a metric in the sense of metric spaces. However, if two curves cross at a point we can use their tangent vectors to define the angle between the curves, so that’s something.

That’s the neatest thing: the metric is defined as a geometric object — a tensor field — so it doesn’t depend on the local coordinate patches at all! All that the patches matter is when you want to represent the inner products with matrices with respect to some basis of the (co)tangent vector space.

[…] want to start getting into a nice, simple, concrete example of the Hodge star. We need an oriented, Riemannian manifold to work with, and for this example we take , which we cover with the usual coordinate […]

[…] some examples will quickly shed some light on this. We can even extend to the pseudo-Riemannian case and pick a coordinate system so that , where . That is, any two are orthogonal, and each either […]

[…] -form, not a vector field, but remember that we’re working in our standard with the standard metric, which lets us use the Hodge star to flip a -form into a -form, and a -form into a vector field! […]

[…] which, though familiar to many, are really heavy-duty equipment. In particular, they rely on the Riemannian structure on . We want to strip this away to find something that works without this assumption, and […]

[…] in hand, we need to properly define the Hodge star in our four-dimensional space, and we need a pseudo-Riemannian metric to do this. Before we were just using the standard , but now that we’re lumping in time we […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.